Octonions

Alberto Elduque

Universidad de Zaragoza

1 Real and complex numbers

2 Quaternions

3 Rotations in three-dimensional space

4 Octonions

2 / 32

1 Real and complex numbers

2 Quaternions

3 Rotations in three-dimensional space

4 Octonions

3 / 32

Real numbers

R = {real numbers}

Real numbers are used in measurements.

0

23

1 2

e π

−1−2

−√

2

But we cannot solve equations as simple as X 2 + 1 = 0!

4 / 32

Real numbers

R = {real numbers}

Real numbers are used in measurements.

0

23

1 2

e π

−1−2

−√

2

But we cannot solve equations as simple as X 2 + 1 = 0!

4 / 32

Real numbers

R = {real numbers}

Real numbers are used in measurements.

0

23

1 2

e π

−1−2

−√

2

But we cannot solve equations as simple as X 2 + 1 = 0!

4 / 32

Complex numbers

C = {a + bi : a, b ∈ R} (' R2)

������������>

4 + 3i

4

3

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

5 / 32

Complex numbers

C = {a + bi : a, b ∈ R} (' R2)

������������>

4 + 3i

4

3

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

5 / 32

Complex numbers

C = {a + bi : a, b ∈ R} (' R2)

������������>

4 + 3i

4

3

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

5 / 32

Complex numbers

C = {a + bi : a, b ∈ R} (' R2)

������������>

4 + 3i

4

3

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

5 / 32

Complex numbers: properties

Exercise

|z1z2| = |z1||z2|

(|.| denotes the usual length.)

Exercise

Rotation of angle α in R2 ↔ multiplication by e iα = cosα+ i sinα.

SO(2) ' {z ∈ C : |z | = 1} ' S1

6 / 32

Complex numbers: properties

Exercise

|z1z2| = |z1||z2|

(|.| denotes the usual length.)

Exercise

Rotation of angle α in R2 ↔ multiplication by e iα = cosα+ i sinα.

SO(2) ' {z ∈ C : |z | = 1} ' S1

6 / 32

Complex numbers: properties

Exercise

|z1z2| = |z1||z2|

(|.| denotes the usual length.)

Exercise

Rotation of angle α in R2 ↔ multiplication by e iα = cosα+ i sinα.

SO(2) ' {z ∈ C : |z | = 1} ' S1

6 / 32

1 Real and complex numbers

2 Quaternions

3 Rotations in three-dimensional space

4 Octonions

7 / 32

A three-dimensional algebra?

Hamilton tried to find a multiplication, analogous to themultiplication of complex numbers, but in dimension 3, that shouldrespect the “law of moduli”: |z1z2| = |z1||z2|:

(a + bi + cj)(a′ + b′i + c ′j) =???

(assuming i2 = −1 = j2)

Problem: ij , ji?

After years of struggle, he found the solution on October 16, 1843.

8 / 32

A three-dimensional algebra?

Hamilton tried to find a multiplication, analogous to themultiplication of complex numbers, but in dimension 3, that shouldrespect the “law of moduli”: |z1z2| = |z1||z2|:

(a + bi + cj)(a′ + b′i + c ′j) =???

(assuming i2 = −1 = j2)

Problem: ij , ji?

After years of struggle, he found the solution on October 16, 1843.

8 / 32

A three-dimensional algebra?

Hamilton tried to find a multiplication, analogous to themultiplication of complex numbers, but in dimension 3, that shouldrespect the “law of moduli”: |z1z2| = |z1||z2|:

(a + bi + cj)(a′ + b′i + c ′j) =???

(assuming i2 = −1 = j2)

Problem: ij , ji?

After years of struggle, he found the solution on October 16, 1843.

8 / 32

A three-dimensional algebra?

Hamilton tried to find a multiplication, analogous to themultiplication of complex numbers, but in dimension 3, that shouldrespect the “law of moduli”: |z1z2| = |z1||z2|:

(a + bi + cj)(a′ + b′i + c ′j) =???

(assuming i2 = −1 = j2)

Problem: ij , ji?

After years of struggle, he found the solution on October 16, 1843.

8 / 32

A three-dimensional algebra?

Hamilton tried to find a multiplication, analogous to themultiplication of complex numbers, but in dimension 3, that shouldrespect the “law of moduli”: |z1z2| = |z1||z2|:

(a + bi + cj)(a′ + b′i + c ′j) =???

(assuming i2 = −1 = j2)

Problem: ij , ji?

After years of struggle, he found the solution on October 16, 1843.

8 / 32

A spark flashed forth

Letter from Sir W. R. Hamilton to his son Rev. Archibald H.Hamilton, dated August 5 1865:

MY DEAR ARCHIBALD -(1) I had been wishing for an occasion of corresponding a littlewith you on QUATERNIONS: and such now presents itself, by yourmentioning in your note of yesterday, received this morning, thatyou “have been reflecting on several points connected with them”(the quaternions), “particularly on the Multiplication of Vectors.”

(2) No more important, or indeed fundamental question, in thewhole Theory of Quaternions, can be proposed than that whichthus inquires What is such MULTIPLICATION? What are itsRules, its Objects, its Results? What Analogies exist between itand other Operations, which have received the same generalName? And finally, what is (if any) its Utility?

9 / 32

A spark flashed forth

(3) If I may be allowed to speak of myself in connexion with thesubject, I might do so in a way which would bring you in, byreferring to an ante-quaternionic time, when you were a merechild, but had caught from me the conception of a Vector, asrepresented by a Triplet: and indeed I happen to be able to put thefinger of memory upon the year and month - October, 1843 - whenhaving recently returned from visits to Cork and Parsonstown,connected with a meeting of the British Association, the desire todiscover the laws of the multiplication referred to regained with mea certain strength and earnestness, which had for years beendormant, but was then on the point of being gratified, and wasoccasionally talked of with you. Every morning in the early part ofthe above-cited month, on my coming down to breakfast, your(then) little brother William Edwin, and yourself, used to ask me,“Well, Papa, can you multiply triplets”? Whereto I was alwaysobliged to reply, with a sad shake of the head: “No, I can only addand subtract them.”

10 / 32

A spark flashed forth

(4) But on the 16th day of the same month - which happened tobe a Monday, and a Council day of the Royal Irish Academy - I waswalking in to attend and preside, and your mother was walkingwith me, along the Royal Canal, to which she had perhaps driven;and although she talked with me now and then, yet anunder-current of thought was going on in my mind, which gave atlast a result, whereof it is not too much to say that I felt at oncethe importance. An electric circuit seemed to close; and a sparkflashed forth, the herald (as I foresaw, immediately) of many longyears to come of definitely directed thought and work, by myself ifspared, and at all events on the part of others, if I should even beallowed to live long enough distinctly to communicate thediscovery.

11 / 32

A spark flashed forth

Nor could I resist the impulse -unphilosophical as it may have been-to cut with a knife on a stone of Brougham Bridge1, as we passedit, the fundamental formula with the symbols, i , j , k; namely,

i2 = j2 = k2 = ijk = −1

which contains the Solution of the Problem, but of course, as aninscription, has long since mouldered away. A more durable noticeremains, however, on the Council Books of the Academy for thatday (October 16th, 1843), which records the fact, that I thenasked for and obtained leave to read a Paper on Quaternions, atthe First General Meeting of the session: which reading took placeaccordingly, on Monday the 13th of the November following.With this quaternion of paragraphs I close this letter I.; but I hopeto follow it up very shortly with another.Your affectionate father, WILLIAM ROWAN HAMILTON.

1The actual name of this bridge is Broome, not Brougham12 / 32

H

H = R1⊕ Ri ⊕ Rj ⊕ Rk ,i2 = j2 = k2 = −1,

ij = −ji = k , jk = −kj = i , ki = −ik = j .

Hamilton and his quaternions

13 / 32

H

H = R1⊕ Ri ⊕ Rj ⊕ Rk ,i2 = j2 = k2 = −1,

ij = −ji = k , jk = −kj = i , ki = −ik = j .

Hamilton and his quaternions

13 / 32

H

H = R1⊕ Ri ⊕ Rj ⊕ Rk ,i2 = j2 = k2 = −1,

ij = −ji = k , jk = −kj = i , ki = −ik = j .

Hamilton and his quaternions

13 / 32

Some properties of H

|q1q2| = |q1||q2| ∀q1, q2 ∈ H(|a + bi + cj + dk|2 = a2 + b2 + c2 + d2)

H is an associative division algebra (but it is notcommutative).Therefore S3 ' {q ∈ H : |q| = 1} is a (Lie) group.(This implies the parallelizability of S3.)

H0 = Ri ⊕ Rj ⊕ Rk ' R3, H = R⊕H0, and ∀u, v ∈ H0:

uv = −u · v + u × v

(where u · v and u × v denote the usual scalar and crossproducts).

14 / 32

Some properties of H

|q1q2| = |q1||q2| ∀q1, q2 ∈ H(|a + bi + cj + dk|2 = a2 + b2 + c2 + d2)

H is an associative division algebra (but it is notcommutative).Therefore S3 ' {q ∈ H : |q| = 1} is a (Lie) group.(This implies the parallelizability of S3.)

H0 = Ri ⊕ Rj ⊕ Rk ' R3, H = R⊕H0, and ∀u, v ∈ H0:

uv = −u · v + u × v

(where u · v and u × v denote the usual scalar and crossproducts).

14 / 32

Some properties of H

|q1q2| = |q1||q2| ∀q1, q2 ∈ H(|a + bi + cj + dk|2 = a2 + b2 + c2 + d2)

H is an associative division algebra (but it is notcommutative).Therefore S3 ' {q ∈ H : |q| = 1} is a (Lie) group.(This implies the parallelizability of S3.)

H0 = Ri ⊕ Rj ⊕ Rk ' R3, H = R⊕H0, and ∀u, v ∈ H0:

uv = −u · v + u × v

(where u · v and u × v denote the usual scalar and crossproducts).

14 / 32

Some properties of H

|q1q2| = |q1||q2| ∀q1, q2 ∈ H(|a + bi + cj + dk|2 = a2 + b2 + c2 + d2)

H is an associative division algebra (but it is notcommutative).Therefore S3 ' {q ∈ H : |q| = 1} is a (Lie) group.(This implies the parallelizability of S3.)

H0 = Ri ⊕ Rj ⊕ Rk ' R3, H = R⊕H0, and ∀u, v ∈ H0:

uv = −u · v + u × v

(where u · v and u × v denote the usual scalar and crossproducts).

14 / 32

Some properties of H

∀q = a1 + u ∈ H, q2 = (a2 − u · u) + 2au, so

q2 − (2a)q + |q|2 = 0 (H is quadratic.)

The map q = a + u 7→ q = a− u is an involution, withq + q = 2a and qq = qq = |q|2.

H = C⊕ Cj ' C2 is a two-dimensional vector space over C.Multiplication is given by:

(p1 + p2j)(q1 + q2j) = (p1q1 − q2p2) + (q2p1 + p2q1)j

15 / 32

Some properties of H

∀q = a1 + u ∈ H, q2 = (a2 − u · u) + 2au, so

q2 − (2a)q + |q|2 = 0 (H is quadratic.)

The map q = a + u 7→ q = a− u is an involution, withq + q = 2a and qq = qq = |q|2.

H = C⊕ Cj ' C2 is a two-dimensional vector space over C.Multiplication is given by:

(p1 + p2j)(q1 + q2j) = (p1q1 − q2p2) + (q2p1 + p2q1)j

15 / 32

Some properties of H

∀q = a1 + u ∈ H, q2 = (a2 − u · u) + 2au, so

q2 − (2a)q + |q|2 = 0 (H is quadratic.)

The map q = a + u 7→ q = a− u is an involution, withq + q = 2a and qq = qq = |q|2.

H = C⊕ Cj ' C2 is a two-dimensional vector space over C.Multiplication is given by:

(p1 + p2j)(q1 + q2j) = (p1q1 − q2p2) + (q2p1 + p2q1)j

15 / 32

1 Real and complex numbers

2 Quaternions

3 Rotations in three-dimensional space

4 Octonions

16 / 32

Rotations in three-dimensional space

q ∈ H, |q| = 1 ⇒ ∃α ∈ [0, π], u ∈ H0, |u| = 1

such that q = (cosα)1 + (sinα)u

Take v ∈ H0 of norm 1 and orthogonal to u, so that {u, v , u × v}is a positively oriented orthonormal basis of R3 = H0.

Consider the linear map:

ϕq : H0 −→ H0,

x 7→ qxq−1 = qxq.

17 / 32

Rotations in three-dimensional space

q ∈ H, |q| = 1 ⇒ ∃α ∈ [0, π], u ∈ H0, |u| = 1

such that q = (cosα)1 + (sinα)u

Take v ∈ H0 of norm 1 and orthogonal to u, so that {u, v , u × v}is a positively oriented orthonormal basis of R3 = H0.

Consider the linear map:

ϕq : H0 −→ H0,

x 7→ qxq−1 = qxq.

17 / 32

Rotations in three-dimensional space

q ∈ H, |q| = 1 ⇒ ∃α ∈ [0, π], u ∈ H0, |u| = 1

such that q = (cosα)1 + (sinα)u

Take v ∈ H0 of norm 1 and orthogonal to u, so that {u, v , u × v}is a positively oriented orthonormal basis of R3 = H0.

Consider the linear map:

ϕq : H0 −→ H0,

x 7→ qxq−1 = qxq.

17 / 32

Rotations in three-dimensional space

q ∈ H, |q| = 1 ⇒ ∃α ∈ [0, π], u ∈ H0, |u| = 1

such that q = (cosα)1 + (sinα)u

Take v ∈ H0 of norm 1 and orthogonal to u, so that {u, v , u × v}is a positively oriented orthonormal basis of R3 = H0.

Consider the linear map:

ϕq : H0 −→ H0,

x 7→ qxq−1 = qxq.

17 / 32

Coordinate matrix of ϕq

ϕq(u) = quq−1 = u (porque uq = qu),

ϕq(v) =((cosα)1 + (sinα)u

)v((cosα)1− (sinα)u)

=((cosα)v + (sinα)u × v

)((cosα)1− (sinα)u)

= (cos2 α)v + 2(cosα sinα)u × v − (sin2 α)(u × v)× u

= (cos 2α)v + (sin 2α)u × v ,

ϕq(u × v) = ... = −(sin 2α)v + (cos 2α)u × v .

18 / 32

Coordinate matrix of ϕq

ϕq(u) = quq−1 = u (porque uq = qu),

ϕq(v) =((cosα)1 + (sinα)u

)v((cosα)1− (sinα)u)

=((cosα)v + (sinα)u × v

)((cosα)1− (sinα)u)

= (cos2 α)v + 2(cosα sinα)u × v − (sin2 α)(u × v)× u

= (cos 2α)v + (sin 2α)u × v ,

ϕq(u × v) = ... = −(sin 2α)v + (cos 2α)u × v .

18 / 32

Coordinate matrix of ϕq

ϕq(u) = quq−1 = u (porque uq = qu),

ϕq(v) =((cosα)1 + (sinα)u

)v((cosα)1− (sinα)u)

=((cosα)v + (sinα)u × v

)((cosα)1− (sinα)u)

= (cos2 α)v + 2(cosα sinα)u × v − (sin2 α)(u × v)× u

= (cos 2α)v + (sin 2α)u × v ,

ϕq(u × v) = ... = −(sin 2α)v + (cos 2α)u × v .

18 / 32

Coordinate matrix of ϕq

ϕq(u) = quq−1 = u (porque uq = qu),

ϕq(v) =((cosα)1 + (sinα)u

)v((cosα)1− (sinα)u)

=((cosα)v + (sinα)u × v

)((cosα)1− (sinα)u)

= (cos2 α)v + 2(cosα sinα)u × v − (sin2 α)(u × v)× u

= (cos 2α)v + (sin 2α)u × v ,

ϕq(u × v) = ... = −(sin 2α)v + (cos 2α)u × v .

18 / 32

Coordinate matrix of ϕq

Thus the coordinate matrix of ϕq relative to the basis {u, v , u × v}is 1 0 0

0 cos 2α − sin 2α0 sin 2α cos 2α

ϕq is a rotation around the axis R+u of angle 2α

19 / 32

Coordinate matrix of ϕq

Thus the coordinate matrix of ϕq relative to the basis {u, v , u × v}is 1 0 0

0 cos 2α − sin 2α0 sin 2α cos 2α

ϕq is a rotation around the axis R+u of angle 2α

19 / 32

Coordinate matrix of ϕq

Thus the coordinate matrix of ϕq relative to the basis {u, v , u × v}is 1 0 0

0 cos 2α − sin 2α0 sin 2α cos 2α

ϕq is a rotation around the axis R+u of angle 2α

19 / 32

SO(3)

The map

ϕ : S3 ' {q ∈ H : |q| = 1} −→ SO(3),

q 7→ ϕq

is a surjective (Lie) group homomorphism with kerϕ = {±1}:

S3/{±1} ' SO(3)

(S3 is the universal cover of SO(3))

20 / 32

SO(3)

The map

ϕ : S3 ' {q ∈ H : |q| = 1} −→ SO(3),

q 7→ ϕq

is a surjective (Lie) group homomorphism with kerϕ = {±1}:

S3/{±1} ' SO(3)

(S3 is the universal cover of SO(3))

20 / 32

SO(3)

The map

ϕ : S3 ' {q ∈ H : |q| = 1} −→ SO(3),

q 7→ ϕq

is a surjective (Lie) group homomorphism with kerϕ = {±1}:

S3/{±1} ' SO(3)

(S3 is the universal cover of SO(3))

20 / 32

SO(3)

The map

ϕ : S3 ' {q ∈ H : |q| = 1} −→ SO(3),

q 7→ ϕq

is a surjective (Lie) group homomorphism with kerϕ = {±1}:

S3/{±1} ' SO(3)

(S3 is the universal cover of SO(3))

20 / 32

Rotations in three-dimensional space

Rotations in R3 ←→ Conjugation in H0 by norm 1

quaternions “modulo ±1”

It is quite easy now to compose rotations in three-dimensionalspace!

It is enough to multiply norm 1 quaternions! (ϕp ◦ ϕq = ϕpq)

Now one can deduce easily the formulas by Olinde Rodrigues(1840) for the composition of rotations.

21 / 32

Rotations in three-dimensional space

Rotations in R3 ←→ Conjugation in H0 by norm 1

quaternions “modulo ±1”

It is quite easy now to compose rotations in three-dimensionalspace!

It is enough to multiply norm 1 quaternions! (ϕp ◦ ϕq = ϕpq)

Now one can deduce easily the formulas by Olinde Rodrigues(1840) for the composition of rotations.

21 / 32

Rotations in three-dimensional space

Rotations in R3 ←→ Conjugation in H0 by norm 1

quaternions “modulo ±1”

It is quite easy now to compose rotations in three-dimensionalspace!

It is enough to multiply norm 1 quaternions! (ϕp ◦ ϕq = ϕpq)

Now one can deduce easily the formulas by Olinde Rodrigues(1840) for the composition of rotations.

21 / 32

Rotations in three-dimensional space

Rotations in R3 ←→ Conjugation in H0 by norm 1

quaternions “modulo ±1”

It is quite easy now to compose rotations in three-dimensionalspace!

It is enough to multiply norm 1 quaternions! (ϕp ◦ ϕq = ϕpq)

Now one can deduce easily the formulas by Olinde Rodrigues(1840) for the composition of rotations.

21 / 32

Rotations in R4

∀p ∈ H with |p| = 1, the left (resp. right) multiplication Lp(resp. Rp) by p is an isometry, due to the multiplicativity ofthe norm.

For p = (cosα)1 + (sinα)u, (α ∈ [0, π], u ∈ H0, |u| = 1), wehave p2 − 2(cosα)p + 1 = 0, so the minimal polynomial ofthe multiplication by p is either X ± 1 for p = ∓1, or theirreducible polynomial X 2 − 2(cosα)X + 1 otherwise.

Hence the characteristic polynomial of the multiplication by pis always (

X 2 − 2(cosα)X + 1)2

and, in particular, the determinant of the multiplication by pis 1.

Multiplication by norm 1 quaternions are rotations in H ' R4.

22 / 32

Rotations in R4

∀p ∈ H with |p| = 1, the left (resp. right) multiplication Lp(resp. Rp) by p is an isometry, due to the multiplicativity ofthe norm.

For p = (cosα)1 + (sinα)u, (α ∈ [0, π], u ∈ H0, |u| = 1), wehave p2 − 2(cosα)p + 1 = 0, so the minimal polynomial ofthe multiplication by p is either X ± 1 for p = ∓1, or theirreducible polynomial X 2 − 2(cosα)X + 1 otherwise.

Hence the characteristic polynomial of the multiplication by pis always (

X 2 − 2(cosα)X + 1)2

and, in particular, the determinant of the multiplication by pis 1.

Multiplication by norm 1 quaternions are rotations in H ' R4.

22 / 32

Rotations in R4

∀p ∈ H with |p| = 1, the left (resp. right) multiplication Lp(resp. Rp) by p is an isometry, due to the multiplicativity ofthe norm.

For p = (cosα)1 + (sinα)u, (α ∈ [0, π], u ∈ H0, |u| = 1), wehave p2 − 2(cosα)p + 1 = 0, so the minimal polynomial ofthe multiplication by p is either X ± 1 for p = ∓1, or theirreducible polynomial X 2 − 2(cosα)X + 1 otherwise.

Hence the characteristic polynomial of the multiplication by pis always (

X 2 − 2(cosα)X + 1)2

and, in particular, the determinant of the multiplication by pis 1.

Multiplication by norm 1 quaternions are rotations in H ' R4.

22 / 32

Rotations in R4

∀p ∈ H with |p| = 1, the left (resp. right) multiplication Lp(resp. Rp) by p is an isometry, due to the multiplicativity ofthe norm.

For p = (cosα)1 + (sinα)u, (α ∈ [0, π], u ∈ H0, |u| = 1), wehave p2 − 2(cosα)p + 1 = 0, so the minimal polynomial ofthe multiplication by p is either X ± 1 for p = ∓1, or theirreducible polynomial X 2 − 2(cosα)X + 1 otherwise.

Hence the characteristic polynomial of the multiplication by pis always (

X 2 − 2(cosα)X + 1)2

and, in particular, the determinant of the multiplication by pis 1.

Multiplication by norm 1 quaternions are rotations in H ' R4.

22 / 32

Rotations in R4

∀p ∈ H with |p| = 1, the left (resp. right) multiplication Lp(resp. Rp) by p is an isometry, due to the multiplicativity ofthe norm.

For p = (cosα)1 + (sinα)u, (α ∈ [0, π], u ∈ H0, |u| = 1), wehave p2 − 2(cosα)p + 1 = 0, so the minimal polynomial ofthe multiplication by p is either X ± 1 for p = ∓1, or theirreducible polynomial X 2 − 2(cosα)X + 1 otherwise.

Hence the characteristic polynomial of the multiplication by pis always (

X 2 − 2(cosα)X + 1)2

and, in particular, the determinant of the multiplication by pis 1.

Multiplication by norm 1 quaternions are rotations in H ' R4.

22 / 32

Rotations in R4

∀p ∈ H with |p| = 1, the left (resp. right) multiplication Lp(resp. Rp) by p is an isometry, due to the multiplicativity ofthe norm.

For p = (cosα)1 + (sinα)u, (α ∈ [0, π], u ∈ H0, |u| = 1), wehave p2 − 2(cosα)p + 1 = 0, so the minimal polynomial ofthe multiplication by p is either X ± 1 for p = ∓1, or theirreducible polynomial X 2 − 2(cosα)X + 1 otherwise.

Hence the characteristic polynomial of the multiplication by pis always (

X 2 − 2(cosα)X + 1)2

and, in particular, the determinant of the multiplication by pis 1.

Multiplication by norm 1 quaternions are rotations in H ' R4.

22 / 32

Rotations in R4

If ψ is a rotation in R4 ' H, a = ψ(1) is a norm 1 quaternion,and

La ◦ ψ(1) = aa = |a|2 = 1,

so La ◦ ψ is actually a rotation in R3 ' H0.

Therefore, there is a norm 1 quaternion q ∈ H such that

aψ(x) = qxq−1

for any x ∈ H. That is:

ψ(x) = (aq)xq−1 ∀x ∈ H.

23 / 32

Rotations in R4

If ψ is a rotation in R4 ' H, a = ψ(1) is a norm 1 quaternion,and

La ◦ ψ(1) = aa = |a|2 = 1,

so La ◦ ψ is actually a rotation in R3 ' H0.

Therefore, there is a norm 1 quaternion q ∈ H such that

aψ(x) = qxq−1

for any x ∈ H. That is:

ψ(x) = (aq)xq−1 ∀x ∈ H.

23 / 32

Rotations in R4

If ψ is a rotation in R4 ' H, a = ψ(1) is a norm 1 quaternion,and

La ◦ ψ(1) = aa = |a|2 = 1,

so La ◦ ψ is actually a rotation in R3 ' H0.

Therefore, there is a norm 1 quaternion q ∈ H such that

aψ(x) = qxq−1

for any x ∈ H. That is:

ψ(x) = (aq)xq−1 ∀x ∈ H.

23 / 32

SO(4)

The map

Ψ : S3 × S3 −→ SO(4),

(p, q) 7→ ψp,q (x 7→ pxq−1)

is a surjective (Lie) group homomorphism with ker Ψ = {±(1, 1)}.

S3 × S3/{±(1,1)} ' SO(4)

(From here we get SO(3)× SO(3) ' PSO(4))

24 / 32

SO(4)

The map

Ψ : S3 × S3 −→ SO(4),

(p, q) 7→ ψp,q (x 7→ pxq−1)

is a surjective (Lie) group homomorphism with ker Ψ = {±(1, 1)}.

S3 × S3/{±(1,1)} ' SO(4)

(From here we get SO(3)× SO(3) ' PSO(4))

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SO(4)

The map

Ψ : S3 × S3 −→ SO(4),

(p, q) 7→ ψp,q (x 7→ pxq−1)

is a surjective (Lie) group homomorphism with ker Ψ = {±(1, 1)}.

S3 × S3/{±(1,1)} ' SO(4)

(From here we get SO(3)× SO(3) ' PSO(4))

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SO(4)

The map

Ψ : S3 × S3 −→ SO(4),

(p, q) 7→ ψp,q (x 7→ pxq−1)

is a surjective (Lie) group homomorphism with ker Ψ = {±(1, 1)}.

S3 × S3/{±(1,1)} ' SO(4)

(From here we get SO(3)× SO(3) ' PSO(4))

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Rotations in R4

It is quite easy to compose rotations in four-dimensional space!

It is enough to multiply pairs of norm 1 quaternions!(ψp1,q1 ◦ ψp2,q2 = ψp1p2,q1q2)

Exercise

What kind of rotation is ψp,q for p + p = 2 cosα andq + q = 2 cosβ?

Solution: A “double rotation” with angles α + β and α− β.

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Rotations in R4

It is quite easy to compose rotations in four-dimensional space!

It is enough to multiply pairs of norm 1 quaternions!(ψp1,q1 ◦ ψp2,q2 = ψp1p2,q1q2)

Exercise

What kind of rotation is ψp,q for p + p = 2 cosα andq + q = 2 cosβ?

Solution: A “double rotation” with angles α + β and α− β.

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Rotations in R4

It is quite easy to compose rotations in four-dimensional space!

It is enough to multiply pairs of norm 1 quaternions!(ψp1,q1 ◦ ψp2,q2 = ψp1p2,q1q2)

Exercise

What kind of rotation is ψp,q for p + p = 2 cosα andq + q = 2 cosβ?

Solution: A “double rotation” with angles α + β and α− β.

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Rotations in R4

It is quite easy to compose rotations in four-dimensional space!

It is enough to multiply pairs of norm 1 quaternions!(ψp1,q1 ◦ ψp2,q2 = ψp1p2,q1q2)

Exercise

What kind of rotation is ψp,q for p + p = 2 cosα andq + q = 2 cosβ?

Solution: A “double rotation” with angles α + β and α− β.

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1 Real and complex numbers

2 Quaternions

3 Rotations in three-dimensional space

4 Octonions

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Octonions (1843-1845)

There is still something in the system which gravelsme. I have not yet any clear views as to the extent towhich we are at liberty arbitrarily to create imaginaries,and to endow them with supernatural properties.

If with your alchemy you can make three pounds ofgold, why should you stop there?

(Letter from John T. Graves to Hamilton, dated October 26,1843!)

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Octonions (1843-1845)

The algebra of quaternions is obtainedby doubling suitably the field ofcomplex numbers:

H = C⊕ Cj .

Doubling again we get the octonions(Graves – Cayley):

O = H⊕Hl .

Arthur Cayley

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Octonions (1843-1845)

The algebra of quaternions is obtainedby doubling suitably the field ofcomplex numbers:

H = C⊕ Cj .

Doubling again we get the octonions(Graves – Cayley):

O = H⊕Hl .

Arthur Cayley

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Octonions (1843-1845)

The algebra of quaternions is obtainedby doubling suitably the field ofcomplex numbers:

H = C⊕ Cj .

Doubling again we get the octonions(Graves – Cayley):

O = H⊕Hl .

Arthur Cayley

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Octonions (1843-1845)

The algebra of quaternions is obtainedby doubling suitably the field ofcomplex numbers:

H = C⊕ Cj .

Doubling again we get the octonions(Graves – Cayley):

O = H⊕Hl .

Arthur Cayley

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Octonions

O = H⊕Hl = R〈1, i , j , k , l , il , jl , kl〉

with multiplication

(p1 + p2l)(q1 + q2l) = (p1q1 − q2p2) + (q2p1 + p2q1)l

and norm:|p1 + p2l |2 = |p1|2 + |p2|2

These are the same formulas that allow us to pass from C to H!

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Octonions

O = H⊕Hl = R〈1, i , j , k , l , il , jl , kl〉

with multiplication

(p1 + p2l)(q1 + q2l) = (p1q1 − q2p2) + (q2p1 + p2q1)l

and norm:|p1 + p2l |2 = |p1|2 + |p2|2

These are the same formulas that allow us to pass from C to H!

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Octonions

O = H⊕Hl = R〈1, i , j , k , l , il , jl , kl〉

with multiplication

(p1 + p2l)(q1 + q2l) = (p1q1 − q2p2) + (q2p1 + p2q1)l

and norm:|p1 + p2l |2 = |p1|2 + |p2|2

These are the same formulas that allow us to pass from C to H!

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Octonions

O = H⊕Hl = R〈1, i , j , k , l , il , jl , kl〉

with multiplication

(p1 + p2l)(q1 + q2l) = (p1q1 − q2p2) + (q2p1 + p2q1)l

and norm:|p1 + p2l |2 = |p1|2 + |p2|2

These are the same formulas that allow us to pass from C to H!

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Octonions

O = H⊕Hl = R〈1, i , j , k , l , il , jl , kl〉

with multiplication

(p1 + p2l)(q1 + q2l) = (p1q1 − q2p2) + (q2p1 + p2q1)l

and norm:|p1 + p2l |2 = |p1|2 + |p2|2

These are the same formulas that allow us to pass from C to H!

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Some algebraic properties

|xy | = |x ||y |, ∀x , y ∈ O.

O is a division algebra, it is neither commutative norassociative!But it is alternative: any two elements generate an associativesubalgebra.Theorem (Zorn 1933): The only finite-dimensional realalternative division algebras are R, C, H and O.

The only such associative algebras R, C and H (Frobenius1877).

S7 ' {x ∈ O : |x | = 1} is not a group (associativity fails), butit constitutes the most important example of a Moufang loop.

O0 = R〈i , j , k , l , il , jl , kl〉. ∀u, v ∈ O0:

uv = −u · v + u × v .

(Cross product in R7!: (u × v)× v = (u · v)v − (v · v)u.)

O is quadratic: ∀x = a1 + u ∈ O, x2 − 2ax + |x |2 = 0.30 / 32

Some geometric properties

The groups Spin7 and Spin8 (universal covers of SO(7) andSO(8)) can be described easily in terms of octonions.

O division algebra ⇒ S7 parallelizable.S1, S3 and S7 are the only parallelizable spheres (Milnor andKervaire).

S6 ' {x ∈ O0 : |x | = 1} is endowed with an almost complexstructure, inherited from the multiplication of octonions.S2 and S6 are the only spheres with such structures (Adams).

Non-desarguesian projective plane OP2.

The only spheres that can be described as homogeneousspaces of nonclassical groups are S6 = AutO/SU(3),S7 = Spin7/AutO and S15 = Spin9/Spin7.

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O is certainly a beautiful mathematical object!

The saying that God is the mathematician, so that,even with meager experimental support, amathematically beautiful theory will ultimately have agreater chance of being correct, has been attributed toDirac. Octonion algebra may surely be called a beautifulmathematical entity.

It is possible that this and other non-associativealgebras (other than Lie algebras) may play someessential future role in the ultimate theory, yet to bediscovered.

(Susumu Okubo)

Thanks

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O is certainly a beautiful mathematical object!

The saying that God is the mathematician, so that,even with meager experimental support, amathematically beautiful theory will ultimately have agreater chance of being correct, has been attributed toDirac. Octonion algebra may surely be called a beautifulmathematical entity.

It is possible that this and other non-associativealgebras (other than Lie algebras) may play someessential future role in the ultimate theory, yet to bediscovered.

(Susumu Okubo)

Thanks

32 / 32

O is certainly a beautiful mathematical object!

The saying that God is the mathematician, so that,even with meager experimental support, amathematically beautiful theory will ultimately have agreater chance of being correct, has been attributed toDirac. Octonion algebra may surely be called a beautifulmathematical entity.

It is possible that this and other non-associativealgebras (other than Lie algebras) may play someessential future role in the ultimate theory, yet to bediscovered.

(Susumu Okubo)

Thanks

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